{
  "id": "2110.04278",
  "version": "v2",
  "published": "2021-10-06T04:34:05.000Z",
  "updated": "2022-03-12T01:08:55.000Z",
  "title": "On Large Values of $|ζ(σ+{\\rm i}t)|$",
  "authors": [
    "Zikang Dong",
    "Bin Wei"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We investigate the extreme values of the Riemann zeta function $\\zeta(s)$. On the 1-line, we obtain a lower bound evaluation $$\\max_{t\\in[1,T]}|\\zeta(1+\\i t)|\\ge {\\rm e}^\\gamma(\\log_2T+\\log_3T+c),$$ with an effective constant $c$ which improves the result of Aistleitner, Mahatab and Munsch. In the half-critical strip $1/2<\\re s<1$, we get an improved $c(\\sigma)$ in the evaluation $$\\max_{t\\in[0,T]}\\log|\\zeta(\\sigma+\\i t)|\\ge c(\\sigma)\\frac{(\\log T)^{1-\\sigma}}{(\\log_2T)^\\sigma},$$ when $\\sigma\\searrow 1/2$, based on an improved lower bound of GCD sums. This improves the result of Bondarenko and Seip.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-12T01:08:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large values",
      "riemann zeta function",
      "lower bound evaluation",
      "gcd sums",
      "extreme values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}